Elliptic partial differential equations perturbed by spatial noise

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1 A stocastic modeling metodology based on weigted Wiener caos and Malliavin calculus Xiaoliang Wan a, Boris Rozovskii b, and George Em Karniadakis b,1 a Program in Applied and Computational Matematics, Princeton University, Princeton, NJ 8544; and b Division of Applied Matematics, Brown University, Providence, RI 2912; Edited by George C. Papanicolaou, Stanford University, Stanford, CA, and approved June 16, 29 (received for review Marc 4, 29) In many stocastic partial differential equations (SPDEs) involving random coefficients, modeling te randomness by spatial wite noise may lead to ill-posed problems. Here we consider an elliptic problem wit spatial Gaussian coefficients and present a metodology tat resolves tis issue. It is based on stocastic convolution implemented via generalized Malliavin operators in conjunction wit weigted Wiener spaces tat ensure te ellipticity condition. We present teoretical and numerical results tat demonstrate te fast convergence of te metod in te proper norm. Our approac is general and can be extended to oter SPDEs and oter types of multiplicative noise. stocastic elliptic problem SPDEs uncertainty quantification Elliptic partial differential equations perturbed by spatial noise provide an important stocastic model in applications, suc as diffusion troug eterogeneous random media (1), stationary Scrodinger equation wit a stocastic potential (2), etc. A typical model of interest is te following stocastic partial differential equation (SPDE): (a(x, ω) u(x)) = f (x), x D R d, u D = g(x), [1] were ω indicates randomness. Recently, tis model as been actively investigated in te context of uncertainty quantification (UQ) for matematical and computational models, see, e.g., refs. 3 and 4 and te references terein. Te diffusion coefficient a(x, ω) is typically modeled by te Karunen Loève type expansion a(x, ω) = ā(x) + ε(x). Here, ā(x) is te mean and ε(x) = k 1 σ k(x) k is te noise term, were σ k (x) are deterministic matrices and := { i } i 1 is a set of uncorrelated random variables wit zero mean and unit variance. For te problem of Eq. 1 to be well posed, te ellipticity condition is required. However, if te noise ε(x) is Gaussian, (or any oter type wit infinite negative part), te standard ellipticity condition would not old, no matter ow small te variance of ε(x, ω) is. Various modifications of te aforementioned setting ave been used to mitigate tis problem. For example, Gaussian models were replaced by distributions wit finite range (e.g., uniform distribution). We note tat from te statistical and, by implication, UQ perspectives, Gaussian perturbations are of paramount importance and sould be modeled judiciously. In tis article, we propose a well-posed modification of te model of Eq. 1 wit Gaussian noise ε = ε(x). More specifically, we replace Eq. 1 by te following SPDE: (ā(x) u(x)) + (δ ε(x) ( u(x))) f (x), x D, u D = g(x), [2] were δ ε(x) denotes te Malliavin divergence operator (MDO) wit respect to Gaussian noise ε(x) (see, e.g., refs. 5 and 6 and te next section). Te MDO δ ε (ζ) is a stocastic integral and specifically, in te present setting, a convolution of te integrand ζ wit te driving Gaussian noise ε. Altoug replacing te ordinary product by stocastic convolution may be surprising at first, it sould be appreciated tat pysical systems rarely produce an instant response to sarp inputs suc as multiplicative forcing. Terefore, in modeling systems subject to sarp perturbations, convolutions often become a model of coice. Historically, stocastic convolution-based models were used to bypass te exceeding singularity of models wit (literally understood) multiplicative noise. In fact, te idea of reduction to MDO could be traced back to te pioneering work of K. Itô on stocastic calculus and stocastic differential equations. Specifically, in is seminal work (7), Itô as replaced te product model u(t) = a(u(t)) + b(u(t)) Ẇ (t) [3] by te stocastic convolution model u(t) = u + t a(u(s))ds + t b(u(s))dw (s), [4] were t b(u(s))dw (s) is te famous Itô s stocastic integral. In tis article, we extend Itô s idea to te infinite dimensional stationary system in Eq. 1 and replace it wit Eq. 2. In contrast to Itô s SDEs, elliptic equations like 1 or 2 are not causal. By tis reason, we replace Itô s integral by te MDO δẇ. Te latter, in tis instance, is equivalent to anticipating (noncausal) Skorood integral, (see ref. 6). A general teory of bilinear elliptic SPDEs tat includes, in particular, Eq. 2 was developed recently in ref. 8. In particular, it was sown tat to ensure well-posedness of Eq. 2 it suffices to assume positivity of ā(x) rater tan of a(x, ω) =ā(x) + ε(x). Tis approac is based on Malliavin calculus and Wiener caos expansion (WCE) wit respect to Cameron Martin basis (see ref. 9 and next section). Te Cameron Martin basis consists of random variables α, were α = (α 1, α 2,...) is a multiindex wit nonnegative integer entries (see te next section for more detail). Te WCE solution of Eq. 2 is given by te series u(x) = α J u α(x) α, were u α = E[u α ]. One can view te Cameron Martin expansion as a Fourier expansion tat separates randomness from te deterministic backbone of te equation. It was sown in ref. 8 tat te WCE coefficients u α (x) for Eq. 2 verify a lower triangular system of linear deterministic elliptic equations (see Eq. 19 below): We refer to tis system as te (uncertainty) propagator. Under very general assumptions, te propagator is equivalent to SPDE 2 in te same way as te set of coefficients of a Fourier expansion is equivalent to te underlying function. In tis article, we develop a numerical metod for solving Eq. 2 wit ig-order discretization in te pysical space and weigted WCE in te probability space. We conduct a teoretical and numerical investigation of te propagation of te truncation errors and illustrate te results by numerical simulations. In particular, an a priori error estimate is presented to demonstrate te Autor contributions: X.W., B.R., and G.E.K. performed researc and wrote te paper. Te autors declare no conflict of interest. Tis article is a PNAS Direct Submission. 1 To wom correspondence sould be addressed. gk@dam.brown.edu. Tis article contains supporting information online at /DCSupplemental. APPLIED MATHEMATICS / cgi / doi / / pnas Early Edition 1of6

2 convergence of te developed numerical algoritm. We also carry out numerical and teoretical comparisons of te model of Eq. 2 wit direct multiplication model of Eq. 1 for positive (lognormal) coefficient a(ω). To facilitate te teoretical comparison of tese models, we ave developed a generalization of te MDO δ ε (based on Gaussian noise ε) to a divergence operator wit respect to a class of noises including nonlinear transformations of Gaussian noise (e.g., lognormal). In addition to its pysical meaning, tere are oter matematical and computational reasons for employing te convolution model of Eq. 2 instead of te multiplication model of Eq. 1, specifically: (i) Te convolution model is computationally efficient due to te lower triangular (in fact, bidiagonal) structure of te propagator; its computational complexity is linear. One could also formally define an approximate WCE solution for te multiplication model of Eq. 1; owever, its propagator would be a full system wit quadratic computational complexity. (ii) Te variance of WCE solutions for bot models is infinite. However, te blow-up of te convolution model is controllable, in tat te WCE solution can be effectively rescaled by simple weigts r α suc tat α r2 α u2 α < (see Teorem 3). We remark tat te blowup in bot models is inevitable even if te perturbation is 1-dimensional (see example in te next section). Te blowup of te multiplication model of Eq. 1 is muc more severe and could ardly be controlled effectively. In fact, we are not aware of any systematic treatment of Eq. 1. Tere are alternative ways to address Eq. 2 and similar bilinear equations. In particular, one could attack Eq. 2 using Hida s wite-noise infinite dimensional calculus (see refs. 1 and 11). In Hida s approac, convolution is interpreted in terms of te Wick product, an operator closely related to stocastic integrals. Convolution models for SPDEs wit positive (lognormal) and normal random coefficients ave been studied extensively (see, e.g., refs. 2 and 12 14) by means of wite-noise analysis. Te wite-noise approac relies on built-in spaces of stocastic distributions known as Hida and Kondratiev spaces (see, e.g., refs. 15 and 16). Te Malliavin calculus is more flexible, and in applications to SPDEs it allows us to build solution spaces optimal for te equation at and (see, e.g., refs. 8 and 17). In tis article, we take advantage of tis important feature of Malliavin calculus to obtain more powerful numerical approximation scemes and substantially more precise estimates of te convergence rates tan tose reported in literature on wite-noise analysis. Malliavin Calculus and Elliptic SPDEs Let F := (Ω, F, P) be a probability space, were F is te σ-algebra generated by := { i } i 1 and U be a real separable Hilbert space. Given a real separable Hilbert space X, we denote by L 2 (F; X) te Hilbert space of square-integrable F-measurable X-valued random elements f. Wen X = R, we write L 2 (F) instead of L 2 (F; R). A formal series Ẇ = k 1 ku k, were {u k, k 1} is a complete ortonormal basis in U, is called Gaussian wite noise on U. Specifically, for te elliptic SPDE we consider ere te proper spaces are X = H 1 (D) and U = L 2 (D). To explain te exact nature of solution to SPDE 2, we recall te notion of Cameron Martin basis. Let J be te collection of multiindices α wit α = (α 1, α 2,...) so tat α k {, 1, 2,...} and α := k 1 α k <. Forα, β J, we define α + β = (α 1 + β 1, α 2 + β 2,...), a = k 1 α k, and α! = k 1 α k!.byε i we denote te multiindex of lengt 1 and wit te single nonzero entry at position i: i.e., te kt coordinate of ε i is1ifk = i andifk = i. Define te collection of random variables Ξ ={ α, α J } as follows: α = k 1 (H α k ( k )/ α k!), were H n (x) are 1-dimensional Hermite polynomials of order n. Teorem 1. [Cameron Martin (9)] Te set Ξ is an ortonormal basis in L 2 (F): if η L 2 (F) and η α = E[η α ], ten η = α J η α α = α J η αh α ()/ α! and E[η 2 ]= α J η2 α. Next, we introduce te Malliavin derivative (MD) and te MDO. To make tese notions more transparent, we begin wit te simplest setting: we will differentiate and integrate α wit respect to a single normal random variable, e.g. k, te kt coordinate of. Te MD D k and divergence operator δ k are defined, respectively, by D k ( α ):= α k α εk, δ k ( α ):= α k + 1 α+εk. [5] In te literature on quantum pysics (see, e.g., ref. 18) te operators δ k and D k are often called creation and anniilation operators, respectively. As intuition suggests, te MDO of D k ( α ) recovers te integrand α. Indeed, for α k >, α 1 k δ k (D k ( α )) = α. Also, MD and MDO can be extended to L 2 (F; X). For example, for f L 2 (F; X U), δẇ (f ) is defined as te unique element of L 2 (F; X) wit te property E[ϕδẆ (f )] =E[(f, DẆ ϕ) U ] for every ϕ suc tat ϕ L 2 (F) and DẆ ϕ L 2 (F; U) (see, e.g., ref. 17). Before proceeding wit te SPDE 2, let us consider a simple example to sed some ligt on te structure of te spaces witin wic we could expect existence of solutions of tis equation. Example 1. Let u be a solution of equation u = 1 + δ (u), [6] were N (, 1). Simple calculations sow tat u 2 L 2 (F) = u2 n =, were u n = n E[uH n()]/ n!. On te oter and, taking a weigted norm wit weigts r n = (n!) 1/2 2 qn/2, q >, we get u 2 RL 2 (F) := n r 2 n u2 n = n 2 qn = (1 2 q ) 1. [7] Te above example demonstrates tat even simple stationary equations do not ave solutions wit finite second moments. To overcome tis obstacle one sould introduce an appropriate weigted version of te solution space. Clearly, introduction of weigts amounts to rescaling of te stocastic Fourier (Wiener caos) representation of te solution. Below, we discuss briefly te construction of weigted spaces. Let R be a bounded linear operator on L 2 (F) defined by R α = r α α for every α J, were te weigts {r α, α J } are positive numbers. In wat follows, we will identify te operator R wit te corresponding collection {r α, α J }. Te inverse operator R 1 is defined as R 1 α = rα 1 α. Te elements of RL 2 (F; X) can be identified wit a formal series α J f α α, were α J f α 2 X r2 α <. Clearly, RL 2(F; X) is a Hilbert space wit respect to te norm f 2 RL 2 (F;X) := Rf 2 L 2 (F;X). We define te space R 1 L 2 (F; X) as te dual of RL 2 (F; X) relative to te inner product in te space L 2 (R; X). For f RL 2 (F; X) and g R 1 L 2 (F) we define te scalar product f, g := E[(Rf )(R 1 g)] X, [8] were f, g = α J f αg α, wit g = α J g α α. Remark 1. One could readily ceck tat u = 1 + u, i.e., te multiplication version of Eq. 6, is muc more complicated and blows up muc faster tan Eq. 6. To address SPDEs driven by lognormal and oter important types of random perturbations, we need to develop a bilinear symmetric version of MDO wit wite noise Ẇ replaced by an arbitrary nonlinear transformation of Gaussian random variables. To begin wit, assume tat vector consists of a single Gaussian random variable N (, 1). Let m = H m ()/ m!. Define a n-tuple MDO by induction: δ 1 ( m):= δ ( m ) and δ n ( m):= 2of6 / cgi / doi / / pnas Wan et al.

3 δ δ (n 1) ( m ) for n > 1. Let δ n ( m):= δ n ( m)/ n!. It is readily cecked tat (m + n)! δ n ( m) = m+n = 1 H m+n (). [9] m!n! m!n! Now, for := ( 1, 2,...) and multiindices α and β, define δ α ( β ):= δ α k ( βk ( k )). Te formula of Eq. 9 translates k into te multidimensional case as follows: δ α ( β ) = (β + α)!/α!β! β+α. [1] Remark 2. (Wick product) Te Wick product ( ), can be defined as follows: for α, β J, H α () H β () := H α+β (), were H γ () := k 1 H γ k ( k ). It follows from Eq. 1 tat H α () H β () = δ H α()(h β ()). Teorem 3. Tere exists an operator R and a unique solution u RL 2 (F; H 1 (D)) of Eq. 12 suc tat: (i) Te operator R is defined by te weigts r α given by r α = qα α!, wit q α = q α k k, were te numbers q k are cosen so tat te renormalization condition Ck 2 q2 k < 1 [17] k 1 olds, and C k are defined by A 1 M k ˆv H 1 (D) C k ˆv H 1 (D), ˆv H1 (D). Teorem 2. If θ, η RL 2 (F; X), set δ α (θ) := β J θ βδ α ( β ) and δ η (θ) := α J η αδ α (θ). Ten δ η is a bounded linear operator on R L 2 (F; X) and δ η (θ) = δ θ (η). [11] Te proof follows from Eq. 1. Example 2. Let N (, 1) and c is a constant. Let θ = exp( c c 2 2 ) and η = exp(c c2 2 ). Ten, δ η(θ) = 1. Tis relation is important for te comparison of te product of Eq. 1 and convolution of Eq. 2 wit lognormal coefficient a(ω) (see Numerical Results). Elliptic SPDE Model. Let us rewrite Eq. 2 as an SPDE in te sense of Malliavin calculus: { Au(x) + δẇ (Mu(x)) = f (x) on D, [12] u(x) = g(x) on D, were D denotes te pysical domain, Ẇ is wite noise on U = L 2 (D), te Hilbert space of square summable sequences of real numbers, Au(x) := i,j D i (a ij (x)d j u(x)), [13a] (ii) Wit te definition of r α given in (i), u α satisfy r 2 α u α 2 H 1 (D) C2 A f 2 H 1 (D) α! α! (C k q k ) 2α k, k 1 were te constant C A is defined by u H 1 (D) C A f H 1 (D). [18] Because te expectation of te MDO is zero, we ave E[Au + δẇ (Mu)] =AE[u] =f (x), wic is te unperturbed (deterministic) version of elliptic Eq. 1. By substituting te WCE u = α J u α α into Eq. 15 and performing a Galerkin projection in te probability space, we can establis te equivalence between Eqs. 2 and 15 and derive te following uncertainty propagator: Au α + k 1 αk M k u α εk = f α, [19] wic is a system of deterministic partial differential equations (PDEs). In te next section, we will develop an efficient numerical algoritm to solve Eq. 19. APPLIED MATHEMATICS M k u(x) := i,j D i ( σ k ij (x)d j u(x) ), [13b] and Mu = k 1 M ku u k. We note tat u k in L 2 (D) can be identified wit ε k, see Eq. 5. For simplicity, we assume tat g = and tat f is deterministic. We can now rewrite te term δẇ (Mu) as δẇ (Mu) = k 1 δ k (M k u). [14] Everywere below we assume tat: (i) te functions a ij (x) and σij k(x) are measurable and bounded in te closure D of D; and (ii) tere exist positive numbers A 1, A 2 suc tat A 1 y 2 a ij (x)y i y j A 2 y 2 for all x D and y R d. Definition 1. A solution to Eq. 12 is a random element u H 1(D) suc tat te equality Au, v + δẇ (Mu), v = f, v [15] were f H 1 (D) and v R 1 L 2 (F) and DẆ v R 1 L 2 (F; U). [16] Analytical issues related to Eq. 15 ave been investigated in ref. 8. In particular, te following result olds: Numerical Metod We will employ WCE in te probability space and a ig-order finite element discretization in te pysical space. Our metod exploits te recursive structure of te propagator of Eq. 19. Tis remarkable property of te propagator significantly reduces te complexity of te algoritm and provides opportunities for improving numerical efficiency. For example, te coefficients u α, satisfying α = p, can be solved in parallel because tey all rely only on already computed coefficients u β wit β < α and α β =1. For simplicity and witout loss of generality, we consider a 2-dimensional pysical domain D.LetT be a family of triangulations of D wit straigt edges and te maximum size of elements in T. We assume tat te family is regular, in oter words, te minimal angle of all te triangles is bounded from below by a positive constant. We define te finite element space as V K = { v : v F 1 K Pˆp(R) }, and V = { v H 1 (D) :v K V K, K T }, were F K is te mapping function for te element K wic maps te reference element R to element K and Pˆp (R) denotes te set of polynomials of degree up to ˆp over R. We assume tat v ƔD =, v V. Tus, V is an approximation of H 1 (D) based Wan et al. Early Edition 3of6

4 on piecewise polynomials. Tere exist many coices of basis functions on te reference elements, suc as -type finite elements (19), spectral/p elements (2, 21), etc. Let J M, p be a finite dimensional subset of J given by J M, p := {α α N M, α p, p N} and R M, p be a given set of weigts r α, α J M, p. We define V c := f = f α α : f RM, p L 2 (F) <, α J M, p c := f = V 1 f α α : f R 1 < M, p L 2 (F). α J M, p Ten, te stocastic finite element metod (sfem) can be defined as: Find u M, p V V c suc tat M Au M, p ( + δ k Mk u M, p ), v = f, v H 1 (D), [2] H 1(D) for any v V Vc 1 were g 1, g 2 H 1 (D) = E[(Rg 1, R 1 g 2 ) H 1 (D) ]. Note tat we truncate te expansion of te wite noise up to M terms and te WCE up to polynomial order p. Let H = H 1(D), and r α = q α / α! as in Teorem 3. Te main result regarding convergence of te stocastic finite element metod (sfem) is given by te following teorem: Teorem 4. For u RL 2 (F; H) RL 2 (F; H m+1 (D)), te error of approximation of te sfem is given by M, p u u RL2 (F; H) ) C ( m u RL2(F; Hm+1) + ˆq W (1 ˆq) + ˆqp+1, [21] 2 1 ˆq were te constant C is independent of, ˆq = k 1 C2 k q2 k < 1, ˆq W = k>m C2 k q2 k, and C k are constants defined in Teorem 3. We remark tat te 3 main components of te error O( m ), O(ˆq W ), and O(ˆq p+1 ) are due to te finite element discretization, te approximation of wite noise, and truncation of te WCE, respectively. We also note tat spectral convergence is obtained in te weigted norm RL2 (F;H). Next, we present a sketc of te proof wereas all tecnical details can be found in supporting information (SI) Appendix. Te approximation error can be decomposed as u u M, p = (u α û α ) α + u α α, α J M, p α J \J M, p were u α and û α are te coefficients of caos expansions of u and u M, p, respectively. Correspondingly, we obtain M, p u u 2 RL 2 (F;H) u α û α 2 H α 2 RL 2 (F) α J M, p + = I 1 + I 2 α J \J M, p u α 2 H α 2 RL 2 (F) To obtain te error contribution I 1, we need to estimate te finite element approximation error u α û α. Since u α depends on u β wit α β =1, we sould consider te error propagation in te uncertainty propagator. Te key observation is tat for α >, te following equations are satisfied in te weak sense Aû α + Au α + M αk M k û α εk =, v V, M αk M k u α εk =, v H, from wic we can obtain a recursive inequality for te error propagation as u α û α H C inf v V u α v H + M c k u α εk û α εk H. We ten use results from te approximation teory to bound te finite element approximation error. Te error contribution from I 2 is from te truncation of WCE and te approximation of wite noise, wic can be estimated using Teorem 3. More details are given in SI Appendix. Numerical Results We consider te 2-dimensional stocastic elliptic problem { [(E[a](x) + δẇ ( u(x))] =f (x), x D, u(x) =, x D, were E[a](x) denotes te mean field of coefficient and Ẇ (x) te random perturbation. For simplicity, we coose te pysical domain D = (, 1) 2, E[a](x) = 1, and f (x) = 1. To represent te wite noise on L 2 (D), we select te following ortonormal basis 1, m = n = w m,n (x) = 2 cos(mπx), n = 2 cos(nπy), m = 2 cos(mπx) cos(nπy), m, n = 1, 2,...,. Hence, we approximate te wite noise as Ẇ (x) = ˆM m+n= w m,n (x) m,n. For convenience, we use an 1-dimensional index and rewrite te above equation as Ẇ (x) = M w k (x) k. Let u(x) = p α = u α α be te truncated WCE of te solution up to polynomial order p. Te uncertainty propagator takes te form (E[a](x) u α ) M αk (w k (x) u α εk ) = f α, were f α = if α >, because f (x) is assumed to be deterministic, and te operator M k is defined as M k u = (w k (x) u). To coose proper r α for te weigted Wiener caos space, we need to specify te constant C k defined in Teorem 3. For our case, we ave C k = w k (x) L (D)/A 1. Here, w k (x) L (D) is uniformly bounded and A 1 = 1 because E[a](x) = 1. Hence, te weigts can be defined as r α = q α / α! wit q α = qα k k,ifq k is cosen as q k = 1/(k + 1)C k. We now examine te convergence of te sfem. We employ te spectral/p element metod to solve te uncertainty 4of6 / cgi / doi / / pnas Wan et al.

5 were indicates stocastic convolution or ordinary multiplication. We coose f (x) = 1, and we consider first te simple model K I (x, ) = exp(c 1 2 c2 ), were N (, 1) is a normal random variable and c is a constant indicating te degree of perturbation. In particular, K I (x) can be regarded as a simplified version of te lognormal noise eẇ. In addition, we consider a second model K II wit space-dependent lognormal noise of te type [ K II (x, ) = exp c ( cos(πx) 2 + ) 2 cos(2πx) c2 ( cos 2 (πx) + 2 cos 2 (2πx) )], [23] Fig. 1. Convergence of te proposed stocastic finite element metod for te model problem. (A) Weigted L 2 norm of approximate wite noise. (B) Spectral convergence of te stocastic finite element metod wit respect to te weigted norm. M = 21. propagator. Te pysical domain (, 1) 2 is discretized in 64 quadrilateral elements, were 12t-order piecewise polynomials are used in eac element. Numerical tests sow tat errors associated wit te pysical discretization are close to macine accuracy and ence negligible. If we fix te number of random variables in te approximation of wite noise and increase te polynomial order of WCE, te dominant error in te error estimate of Eq. 21 sould be te truncation error of te WCE. In Fig. 1, we plot te convergence beavior of te approximate wite noise in te weigted norm RL 2 (F, L 2 (D)) wit respect to M. We also plot te errors of te approximate solution in te weigted norm RL 2 (F, H 1 (D)) wit respect to te polynomial order of WCE. For te numerical simulation, we coose M = 21, corresponding to m + n 5. Te M, p+1 errors are approximated as u u M, p RL2 (F;H 1 (D)). We see tat spectral convergence is obtained, wic is consistent wit te error estimate of Eq. 21. were i, i = 1, 2, 3, are normal random variables. In oter words, we take te first 3 modes of wite noise Ẇ (x) = 1 + i=2 2 cos((i 1)πx)i on L 2 (, 1). It is easy to sow tat E[K II (x, )] =1, and tat Var(K II (x, )) = exp(c 2 + 2c 2 cos 2 (πx) + 2c 2 cos 2 (2πx)) 1. In Fig. 2, we plot te variances of solutions corresponding to 2 different models obtained from solving bot te convolution and te multiplication cases; results for bot K I and K II are sown. We observe tat wen te noise is small, te variances given by te 2 models are about te same; owever, wen te noise is large a large difference exists, wit te variance given by te multiplication model increasing muc faster tan tat given by te convolution model. For K I, wen its standard deviation increases from.13 to , te solution variance increases from O(1 5 )too(1 6 ) for te multiplication model, and from O(1 5 )too(1 1 ) for te convolution model a 5- order difference in magnitude! For K II te variace acieves larger values as we include more terms in te expansion for te wite noise. More results and analysis for bot models are included in SI Appendix. Tis exponential increase of te variance wit respect to perturbation can be explained by referring to te aforementioned Example 2. Specifically, we consider te simple problem (δ a (u x (x, ))) x = f (x), x (x, x 1 ), u(x ) = u(x 1 ) = wit lognormal but space-independent coefficient a(ω) = exp(c c2 2 ); te corresponding solution is u(x) = θδ 1 f (x) wit δ being te Diriclet Laplacian on (x, x 1 ) and δ a (θ) = 1. Example 2 sows tat θ = exp( c c2 ). On te oter and, te solution to te corresponding multiplication model (a v x (x, )) x = 2 f (x) isv(x, ) = a 1 δ 1 f (x). Hence, te rapid increase in te variance observed in te computations is related to te ratio APPLIED MATHEMATICS Model Comparison. Next, we present a comparison of te convolution and multiplication models by considering te following 1-dimensional problem d (K(x) ddx ) dx u = f (x), [22] u() =, u(1) =, Fig. 2. Variance versus perturbation at x =.5. For model II, it is very expensive to compute te solution beyond c = 1 for te multiplication model. Wan et al. Early Edition 5of6

6 a 1 /θ, wic is equal to exp(c 2 ). Clearly, v(x) = exp(c 2 )u(x) and E[ v(x) k ] = exp(kc 2 )E[ u(x) k ]. Tis establises tat solutions of te above equation wit lognormal permeability corresponding to multiplication model are exceedingly singular as compared to a similar equation wit stocastic convolution based on MDO. Summary and Discussion In tis article, we ave developed an efficient numerical metod for solving second-order elliptic SPDEs wit Gaussian coefficients. We introduced te concept of stocastic convolution model and employed Malliavin calculus to implement it. Te stocastic solution was represented by a weigted WCE in te appropriate norm. It was sown tat te coefficients of te WCE can be obtained by solving a lower triangular system of deterministic elliptic PDEs, i.e., te uncertainty propagator. We derived a a priori error estimate to measure te rate of convergence wit respect to appropriately weigted L 2 norm. We ave also carried out numerical and teoretical comparisons of te model of Eq. 2 wit te direct multiplication model of Eq. 1 for positive (lognormal) coefficient a(ω). To facilitate te teoretical comparison of tese models, we ave developed a generalization of te MDO δ ε (based on Gaussian noise ε) to a divergence operator wit respect to a class of noises including nonlinear transformations of Gaussian noise (e.g., lognormal). It migt be instructive to reexamine te differences and similarities between te convolution model Au(x) = δẇ (Mu(x)) and its multiplication counterpart Au(x) = Mu(x) Ẇ (x). It is well understood tat te multiplication models are exceedingly singular. We are not aware of any systematic teoretical or numerical efforts to investigate suc SPDEs. It appears toug tat te general metodology developed in tis article could be extended to address elliptic equations of tis type as well. However, in te 1. Papanicolaou G (1995) Diffusion in random media. Surveys in Applied Matematics, eds Keller JB, McLauglin D, Papanicolaou G (Plenum, New York). 2. Holden H, Oksendal B, Uboe Zang T (1996) Stocastic Partial Differential Equations: A modeling, Wite Noise Functional Approac (Birkauser, Boston). 3. Babuška I, Tempone R, Zouraris G (24) Galerkin finite element approximations of stocastic elliptic differential equations. SIAM J Numer Anal 42: Frauenfelder P, Scwab C, Todor R (25) Finite elements for elliptic problems wit stocastic coefficients. Comput Metods Appl Mec Eng 194: Malliavin P (1997) Stocastic Analysis (Springer, Berlin). 6. Nualart D (26) Malliavin Calculus and Related Topics (Springer, New York), 2nd Ed. 7. Itô K (1944) Stocastic integral. Proc Imp Acad Tokyo Jpn 2: Lototsky S, Rozovskii B (29) Stocastic partial differential equations driven by purely spatial noise. SIAM J Mat Anal, in press. 9. Cameron R, Martin W (1947) Te ortogonal development of nonlinear functionals in series of Fourier Hermite functionals. Ann Mat 48: Hida T, Juo H, Pottoff J, Streit L (1993) Wite Noise (Kluwer Academic, Dordrect, Te Neterlands). 11. Kuo HH (1966) Wite Noise Distribution Teory (CRC, Boca Raton, FL). multiplication setting, te propagator is not lower triangular, and te solution spaces are expected to be muc larger tan in te setting of tis article. In contrast to multiplication models te convolution models are muc more manageable analytically and numerically and ave solid pysical meaning. Bot models become muc easier if one replaces te wite noise forcing by lognormal (wic is a positive noise). We ave sown tat in te lognormal setting te SPDE as reasonable solutions for bot models. Still, as te variance of te noise grows, te variance of te solution of te multiplication equation scales up exponentially as compared wit te convolution model. In spite of te aforementioned differences, te 2 models are closely related. In fact, te convolution models could be viewed as te igest stocastic order approximations to multiplication models. Indeed, by Remark 2, δ H α()(h β ()) = H α+β () wile H α () H β () = H α+β () + R α,β, were R α,β is a linear combination of Hermite polynomials of orders lesser tan α + β. Finally, we note tat te mean in te multiplication model for te problems considered ere deviate greatly from te corresponding deterministic solution (see SI Appendix). Clearly, linear systems wit additive noise are unbiased perturbations of teir deterministic counterparts wit te statistical average of a solution to randomized system coinciding wit te solution of te original unperturbed system. Te convolution bilinear equations considered in tis article also enjoy tis important property of linear systems. However, we stress tat tis property does not old and sould not be expected from SPDEs wit nonlinear operator A or/and multiplication in te stocastic terms. ACKNOWLEDGMENTS. We acknowledge te elpful suggestions by te referees and te useful discussions wit our students Z. Q. Zang and C.-Y. Lee. Tis work was supported by te National Science Foundation (NSF)/Division of Matematical Sciences and Army Researc Office grants (to B.R.) and NSF/Applied Matematics Crosscutting Stocastic Systems and Office of Naval Researc (G.E.K.). 12. Manouzi H, Teting T (26) Numerical Analysis of te Stocastic Stokes Equations of Wick Type (Wiley InterScience, New York). 13. Teting T (2) Solving Wick-stocastic boundary value problems using a finite element metod. Stocastics Stocastic Rep 7: Våge G (1998) Variational metods for PDEs applied to stocastic partial differential equations. Mat Scand 82: Kondratiev Y, Samoylenko YS (1978) Te spaces of trial and generalized functions of infinite number of variables. Rep Mat Pys 14: Kondratiev Y, Leukert P, Pottoff J, Streit L, Westerkamp W (1966) Generalized functionals in Gaussian spaces: Te caracterization teorem revisited. 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